24 Feb, 2016
| EV | SE | ||
|---|---|---|---|
| Proportions | sum | $p\times n$ | $\sqrt{p(1-p)}\times \sqrt{n}$ |
| mean | $p$ | $\sqrt{p(1-p)} / \sqrt{n}$ | |
| Means | sum | $\bar{x} \times n$ | $\displaystyle\sqrt{\frac{(x_1-\bar{x})^2+…+x_n-\bar{x})^2}{n}} \times \sqrt{n}$ |
| mean | $\bar{x}$ | $\displaystyle\sqrt{\frac{(x_1-\bar{x})^2+…+x_n-\bar{x})^2}{n}} / \sqrt{n}$ |
Note that the Standard Errors (SE) are computed from the standard deviation of the box (one draw from the box).
A confidence interval is computed as follows:
\[\text{CI} = \text{estimate} \pm z^* \times \text{SE}\]So, the $95\%$ confidence interval for a mean is:
\[\bar{x} \pm 1.95 \times \frac{s}{\sqrt{n}}, \text{ where } s = \displaystyle\sqrt{\frac{(x_1-\bar{x})^2+...+x_n-\bar{x})^2}{n}}\]and the $95\%$ condifence intercal for a proportion is:
\[p \pm 1.95 \times \frac{\sqrt{p(1-p)}}{\sqrt{n}}\]